Tensor Networks for Multi-Modal Non-Euclidean Data

Modern data sources are typically of large scale and multi-modal natures, and acquired on irregular domains, which poses serious challenges to traditional deep learning models. These issues are partially mitigated by either extending existing deep learning algorithms to irregular domains through graphs, or by employing tensor methods to alleviate the computational bottlenecks imposed by the Curse of Dimensionality. To simultaneously resolve both these issues, we introduce a novel Multi-Graph Tensor Network (MGTN) framework, which leverages on the desirable properties of graphs, tensors and neural networks in a physically meaningful and compact manner. This equips MGTNs with the ability to exploit local information in irregular data sources at a drastically reduced parameter complexity, and over a range of learning paradigms such as regression, classification and reinforcement learning. The benefits of the MGTN framework, especially its ability to avoid overfitting through the inherent low-rank regularization properties of tensor networks, are demonstrated through its superior performance against competing models in the individual tensor, graph, and neural network domains.

Nonstationary Portfolios: Diversification in the Spectral Domain

Classical portfolio optimization methods typically determine an optimal capital allocation through the implicit, yet critical, assumption of statistical time-invariance. Such models are inadequate for real-world markets as they employ standard time-averaging based estimators which suffer significant information loss if the market observables are non-stationary. To this end, we reformulate the portfolio optimization problem in the spectral domain to cater for the nonstationarity inherent to asset price movements and, in this way, allow for optimal capital allocations to be time-varying. Unlike existing spectral portfolio techniques, the proposed framework employs augmented complex statistics in order to exploit the interactions between the real and imaginary parts of the complex spectral variables, which in turn allows for the modelling of both harmonics and cyclostationarity in the time domain. The advantages of the proposed framework over traditional methods are demonstrated through numerical simulations using real-world price data.

In-Ear SpO2 for Classification of Cognitive Workload

Classification of cognitive workload promises immense benefit in diverse areas ranging from driver safety to augmenting human capability through closed loop brain computer interface. The brain is the most metabolically active organ in the body and increases its metabolic activity and thus oxygen consumption with increasing cognitive demand. In this study, we explore the feasibility of in-ear SpO2 cognitive workload tracking. To this end, we preform cognitive workload assessment in 8 subjects, based on an N-back task, whereby the subjects are asked to count and remember the number of odd numbers displayed on a screen in 5 second windows. The 2 and 3-back tasks lead to either the lowest median absolute SpO2 or largest median decrease in SpO2 in all of the subjects, indicating a robust and measurable decrease in blood oxygen in response to increased cognitive workload. Using features derived from in-ear pulse oximetry, including SpO2, pulse rate and respiration rate, we were able to classify the 4 N-back task categories, over 5 second epochs, with a mean accuracy of 94.2%. Moreover, out of 21 total features, the 9 most important features for classification accuracy were all SpO2 related features. The findings suggest that in-ear SpO2 measurements provide valuable information for classification of cognitive workload over short time windows, which together with the small form factor promises a new avenue for real time cognitive workload tracking.

Multi-Graph Tensor Networks

The irregular and multi-modal nature of numerous modern data sources poses serious challenges for traditional deep learning algorithms. To this end, recent efforts have generalized existing algorithms to irregular domains through graphs, with the aim to gain additional insights from data through the underlying graph topology. At the same time, tensor-based methods have demonstrated promising results in bypassing the bottlenecks imposed by the Curse of Dimensionality. In this paper, we introduce a novel Multi-Graph Tensor Network (MGTN) framework, which exploits both the ability of graphs to handle irregular data sources and the compression properties of tensor networks in a deep learning setting. The potential of the proposed framework is demonstrated through an MGTN based deep Q agent for Foreign Exchange (FOREX) algorithmic trading. By virtue of the MGTN, a FOREX currency graph is leveraged to impose an economically meaningful structure on this demanding task, resulting in a highly superior performance against three competing models and at a drastically lower complexity.

Recurrent Graph Tensor Networks

Recurrent Neural Networks (RNNs) are among the most successful machine learning models for sequence modelling. In this paper, we show that the modelling of hidden states in RNNs can be approximated through a multi-linear graph filter, which describes the directional flow of temporal information. The so derived multi-linear graph filter is then generalized to a tensor network form to improve its modelling power, resulting in a novel Recurrent Graph Tensor Network (RGTN). To validate the expressive power of the derived network, several variants of RGTN models were proposed and employed for the task of time-series forecasting, demonstrating superior properties in terms of convergence, performance, and complexity. By leveraging the multi-modal nature of tensor networks, RGTN models were shown to out-perform a standard RNN by 23% in terms of mean-squared-error while using up to 86% less parameters. Therefore, by combining the expressive power of tensor networks with a suitable graph filter, we show that the proposed RGTN models can out-perform a classical RNN at a drastically lower parameter complexity, especially in the multi-modal setting.

Tensor Decompositions in Deep Learning

The paper surveys the topic of tensor decompositions in modern machine learning applications. It focuses on three active research topics of significant relevance for the community. After a brief review of consolidated works on multi-way data analysis, we consider the use of tensor decompositions in compressing the parameter space of deep learning models. Lastly, we discuss how tensor methods can be leveraged to yield richer adaptive representations of complex data, including structured information. The paper concludes with a discussion on interesting open research challenges.

Supervised Learning for Non-Sequential Data with the Canonical Polyadic Decomposition

There has recently been increasing interest, both theoretical and practical, in utilizing tensor networks for the analysis and design of machine learning systems. In particular, a framework has been proposed that can handle both dense data (e.g., standard regression or classification tasks) and sparse data (e.g., recommender systems), unlike support vector machines and traditional deep learning techniques. Namely, it can be interpreted as applying local feature mappings to the data and, through the outer product operator, modelling all interactions of functions of the features; the corresponding weights are represented as a tensor network for computational tractability. In this paper, we derive efficient prediction and learning algorithms for supervised learning with the Canonical Polyadic (CP) decomposition, including suitable regularization and initialization schemes. We empirically demonstrate that the CP-based model performs at least on par with the existing models based on the Tensor Train (TT) decomposition on standard non-sequential tasks, and better on MovieLens 100K. Furthermore, in contrast to previous works which applied two-dimensional local feature maps to the data, we generalize the framework to handle arbitrarily high-dimensional maps, in order to gain a powerful lever on the expressiveness of the model. In order to enhance its stability and generalization capabilities, we propose a normalized version of the feature maps. Our experiments show that this version leads to dramatic improvements over the unnormalized and/or two-dimensional maps, as well as to performance on non-sequential supervised learning tasks that compares favourably with popular models, including neural networks.